Keywords: Mathematical Modeling, Elementary Education, Teaching Practices, Professional Development, Learning Progressions, Knowledge of Content and Pedagogy

Overview of the Working Group

This is a new working group began in 2017 and builds on PMENA’s long tradition of working groups on Models and Modeling. Our goal is to broaden the access of mathematical modeling to elementary grades and advance the field’s collective understanding of the interrelated processes of mathematical modeling in the elementary grades and beyond. Although there has been a long history of mathematical modeling at PME and PMENA, the focus has primarily been on middle, high school and university levels. We believe it is critically important to understand the learning progression of mathematical modeling from elementary to secondary grades to ensure coherence and rigor in the mathematics curriculum. Implementing mathematical modeling in the elementary grades is not just going “light” with the high school math modeling curriculum. Instead we advocate integrating aspects of mathematical modeling in the early grades effectively to enhance student learning and to help build their competency in real-world problem solving using their current mathematical knowledge. The latter content knowledge is expected to evolve as students continue to learn new mathematics as they progress towards high school and beyond. So what does mathematical modeling look like in the elementary grades?

The working group leaders come from a multi- university collaboration working with school districts (with diverse populations) to understand the nature of mathematical modeling in the elementary grades. In our design-based implementation research, each university site worked with the collaborating district’s teacher leaders to co-plan the professional development. Teachers became co-designers of the mathematical modeling curriculum for the elementary classrooms. In our project, we engaged elementary teachers in considering mathematical modeling using real world tasks that contained several of the following attributes: (a) Open-endedness; (b) Problem-posing; (c) Creativity and choices; d) Iteration and revisions.

Why focus on early grades? In addition to the direct benefits of modeling, the elementary school environment affords many advantages that complement work in mathematical modeling. Elementary students often rely on using concrete referents such as objects, drawings, diagrams, and actions or pictures to help conceptualize and to construct carefully formulated arguments to solve a problem. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades (CCSSO 2010). Young students have high potential to become fluent – native speakers, thinkers and dreamers of mathematics. Thinking creatively may come more easily to children first learning and exploring mathematical concepts. Kindergarten students can use manipulatives to independently solve traditional multiplication or division problems they have never seen before, which is evidence that students come with knowledge--we don’t have to wait to incorporate modeling activities until we have “shown them how” to do everything. Because early grade teachers are generalists, they can address several subjects simultaneously through modeling activities. Mathematical Modeling is of interest and relevance to the mathematics education community especially because it connects to the need for professional development focused on mathematical modeling in the elementary grades.

Our researchers used Design-Based Implementation Research methodology, DBIR (Fishman, Penuel, Allen, Cheng, & Sabelli, 2013) to examine the design of our professional development and to study and enhance our design through feedback from our iterative implementation cycles. DBIR was a method of choice for our study because it has (1) a focus on problems of practice from multiple stakeholders’ perspectives; (2) a commitment to iterative, collaborative design; (3) a concern with developing theory and knowledge related to both classroom learning and capacity for sustaining change in systems (Fishman, Penuel, Allen, Cheng, & Sabelli, 2013, p. 136).

Through our work, we are gaining a better sense of teaching practices and classroom routines that support modeling. We are contributing to the understanding of what is possible in early elementary grades and how these processes support the development of critical 21st century skills. As we continue in our research to consider what constitutes the practice of Mathematical Modeling (MM) and how it could be implemented in classrooms at different grain size, we invite the larger PMENA community to build on this knowledge. Over the past decades, working group leaders have individually and in subgroups, been theorizing about as well as collecting, analyzing, and reporting on data relating to mathematics modeling. This Working Group builds on and extends the work of previous Model and Modeling tradition by discussing current work from leading scholars from diverse perspectives.

Relevance to Psychology of Mathematics Education

The purpose of this working group is to invite individuals across the research community interested in synthesizing the literature and collaborating on research focused on mathematical modeling along the developmental continuum. Our goal of mapping a learning progression of mathematical modeling from K-12 education, particularly starting from elementary to middle grades is critically important to provide coherence in the mathematics curriculum.

The primary focus for this working group will be around the following three goals:

1. Bring together scholars with an interest in examining research with meaningful data consisting of student MM artifacts and teachers’ content and modeling competencies.

2. Map the learning progression for mathematical modeling and task design for K-6 mathematics education and beyond.

3. Begin dialogue and collaboration among individuals and groups conducting research on student- and teacher-related outcomes related to implementing mathematical modeling, ways mathematical modeling promotes 21st century skills, and interdisciplinary skills in STEM.

Related Research

The complexity of the modern world places more demand and importance in developing students’ abilities to deal with demands of our society (e.g. Gravemeijer, Stephan, Julie, Lin, & Ohtani, 2017). These abilities include interdisciplinary problem solving, techno-mathematical literacy, flexibility in applying numerical and algebraic reasoning, thinking critically, and constructing, describing, explaining, manipulating and predicting complex systems (English, 2013). Mathematical modeling (MM) is seen as a powerful tool for advancing students understanding of mathematics and for developing an appreciation of mathematics as a tool for analyzing critical issues in the real-world, that is, the world outside of the mathematics classroom (Greer & Mukhopadhyay, 2012). Traditionally, MM has been implemented primarily in secondary schools, but recent research examines this approach with elementary students to promote their problem solving and problem-posing abilities (e.g. English, 2010). MM provides the opportunity for students to solve genuine problems and to construct significant mathematical ideas and processes instead of simply executing previously taught procedures and is important in helping students understand the real world (English, 2010).

It must be pointed out that the phrases mathematical modeling and modeling mathematics are used in different ways. Cirillo, Pelesko, Felton-Koestler, and Rubel (2016) succinctly describe modeling mathematics as the use of representations to communicate mathematical concepts or ideas. The central characteristic of modeling mathematics is that the process “begins in the mathematical world, rather than in the real world (Cirillo et al., 2016, p. 4). For example, Lesh, Post, and Behr (1987) describe five representations that support students in understanding mathematical concepts or ideas: pictures, manipulatives, written symbols, oral language and real-world situations. For Lesh et al. (1987), the real-world situations provided a context for the problems; the representations began in the mathematical world. A form of mathematical modeling instruction introduced by Lesh & Doerr, 2003, Model-eliciting activities (MEAs), incorporate client-driven, real-life contexts and open-ended problem solving. Mathematical modeling is a process that starts in the real world and makes sense of non-mathematical situations in a mathematical format (English, Fox and Watters, 2005). The mathematical modeling process involves both the creation and the continuous modification of models of empirical situations to both understand them better and enhance decision-making in real-time. As students create and modify mathematical models to understand and solve real-world problems, they engage in a cyclical process of generating and validating their model and results. The figure below illustrates a cycle used with elementary students to help organize reasoning in mathematical modeling by using terms comprehensible to young math modelers.

Figure 1. Proposed Modeling Cycle for Elementary Students (Levy R., Cordeiro J., E. Lane, A. Sierra, Sinclair D., Yang L. & Matson K., 2016

One of the ways, the researchers in this working group approached MM in the elementary grades was to immerse students in a real world situation within their local context that was relatable to and personally meaningful. To keep the initial problem open-ended, students were encouraged to develop the habit of mind of being problem posers by identifying the many questions around the real phenomenon, then defining a mathematical problem that can be solved by way of mathematics. After the identification process of the problem, the modeler makes assumptions, eliminates unnecessary information, and identifies important quantities in order to form the model. This real-world model becomes a mathematical model when the processes are replaced by mathematical symbols, relations and operations. It should be noted that there can be several mathematical models for a given real-world situation. Next, the model is solved mathematically and results are translated back to the real-world and interpreted in the original context. The problem solver then validates the model by checking whether the solution is appropriate or reasonable for the purpose. This process of making assumptions, identifying variables, formulating the model, interpreting the result, and validating the model is iterative in nature and is modified or changed and repeated until a satisfactory solution has been obtained and communicated (Blum, 2002). It is important to note that teachers play a crucial role in MM. The teacher must be able to: (a) provide opportunities for students to acquire mathematical competencies and make connections between the real world and mathematics; (b) maintain the high cognitive demand of the MM process; and (c) provide classroom management that is learner-centered (Blum & Ferri, 2009).

Previous work with elementary school children demonstrated it is feasible for them to develop a disposition towards realistic mathematical modeling (Lieven & De Corte, 1997). One of the issues in implementing MM at the elementary level is that MM can be difficult for both teachers and students to implement (Blum & Ferri, 2009). MM can be difficult for teachers to implement as they must be able to merge mathematical content and real-world applications while teaching in a more open-ended and less predictable way (Blum & Ferri, 2009). It can be a challenge for students because each step of the modeling process presents a possible cognitive barrier (Blum & Ferri, 2009). As stated in the Common Core Standards for Mathematical Modeling, “Real-world situations are not organized and labeled for analysis; formulating tractable models, representing such models, and analyzing them is appropriately a creative process. These real-world problems tend to be messy and require multiple math concepts, a creative approach to math, and involves a cyclical process of revising and analyzing the model” (Carter et. al., 2009).

In a previous PMENA report, Suh, Matson, Williams and Seshaiyer (2016) reported the challenges and affordances of mathematical modeling in the early grades.

Teacher Challenges.The challenges teachers faced when implementing mathematics modeling in the elementary grades included: a) Novelty and ambitious nature of the modeling process-When implementing MM in the classrooms for the first time, teachers found it was difficult to move students through the full process as it was a novel approach and students had never been introduced to creating and validating their mathematical models; b) Managing discourse- Another difficulty encountered by teachers in the MM process was in defining their role as facilitators. The teachers commented that “...it is really difficult as a teacher to help students find a direction to go with their solution but not direct or guide them toward a teacher goal.”; and c) Constraints around mandated standards-Participants acknowledged that MM takes time to implement in the classroom and that additional class time to implement these tasks would be helpful. An additional challenge noted by teachers was that mathematical modeling didn't go the way they expected it to and they wrestled with the need to meet state standards.

Affordances of Mathematical Modeling. The main affordances our teacher-designers mentioned were that mathematical modeling provided opportunity for content to be covered without direct instruction, had interdisciplinary connections, and provided mathematical relevance, and student engagement: a) Content covered without direct instruction- When teachers implemented MM in their classrooms for the first time they were amazed at the amount of content that could be covered without direct instruction. Students could see how the mathematics could serve their needs as they used the mathematics they learned while other times, the mathematics related to future learning objectives which allowed them to revisit their model as their learning progressed; b) Interdisciplinary opportunities-Another positive take-away from implementing MM in these teachers’ classrooms for the first time was how MM created a space where content covered was interdisciplinary connecting to social studies, STEM and language arts; c) Relevance-By providing authentic tasks for students to grapple with through the MM process, mathematics became relevant to the students; d) Student engagement-A number of our teachers indicated how engaged their students were in their MM tasks. Mathematical modeling inspired these teachers’ endeavors and provided pictures of practices that served as the proof of concept they needed to sustain their professional commitment to mathematical modeling.

Support Teachers Need. The three main areas of support teacher-designers requested were access to MM resources, pictures of practice, time and collaboration with like-minded teachers: a) Resources and pictures of practice-Teachers indicated a desire to use MM in their classrooms but indicated a need for a bank of open-ended MM lessons and new ideas for continuing to create these lessons; b) Time-Teachers expressed the need for more time to work through and become comfortable with implementing the modeling process in their classrooms. Teachers noted it was only in working through the MM cycle several times that they felt comfortable with the process and felt their students were able to understand the whole MM process; and c) Teacher collaboration-Teachers indicated a desire to continue to work with a cohort to build MM lessons; to observe other teachers implement MM in their classrooms; and to work alongside a colleague who valued MM and with whom they could share ideas.

Other related research the team will share include, Carlson, Wickstrom, Burroughs & Fulton’s (2016) work, A Case for Mathematical Modeling in the Elementary School Classroom, where they provide a teaching framework for MM using the "organize - monitor - regroup" cycle to support the teachers’ work in engaging young students in modeling. Wickstrom, Carr and Lackey (2007) will showcase an engaging article using mathematical modeling to explore Yellowstone National Park. Suh, Matson, & Seshaiyer (2017) will also share ways in which mathematical modeling enhanced students creativity, collaboration, critical thinking and communication skills and exposed students to interdisciplinary themes of service learning and STEM integration.

Plan for Active Engagement of Participants

The working group will meet three times during the conference and virtually during the course of one year. In each session, PMENA members will engage in mathematical modeling while sharing their perspectives in teaching and learning mathematics, considering synergistic areas fruitful for future research and practice, and finding collaborators within our group.

Session 1: Exploring the Nature of Mathematical Modeling in the Early Grades

The first session will focus on better understanding the nature of mathematical modeling in the elementary grades while considering the student perspective and recognizing the importance of teachers knowing their students and the contexts that are meaningful to their students. We will examine how mathematical modeling used by K-6 teachers demonstrates the interdisciplinary nature of mathematical modeling, the diversity of mathematical approaches taken by student modelers, and the multiple pathways the teacher can use to elicit students’ mathematical thinking. Exemplar tasks that emphasized local contexts and tapped into students’ funds of knowledge and student artifacts will be shared to illustrate the child’s perspective and the developmental progression. These topics will facilitate group discussions exploring the learning progression for mathematical modeling thinking and habits of mind that can develop for emergent mathematical modelers from an early grade.

Session 2: Identifying the Knowledge of Content and Pedagogy Needed for Mathematical Modeling in the Elementary Grades

In our second session, we will focus on clearly defining modeling teaching practices and competencies needed for mathematical modeling and outlining research goals and objectives to monitor the enactment of these practices. We will detail classroom routines, such as the "organize - monitor - regroup" cycle (Carlson, et al. 2017), and the Pedagogical Practices for Mathematical Modeling (Suh, Matson, & Seshaiyer, in press) as we share designed activities and lesson vignettes to solicit more ideas around high leverage MM teaching practices. We will explore what mathematical knowledge is needed to “successfully” facilitate mathematical modeling tasks in elementary grades.

Session 3: Finding the Synergy between Mathematical Modeling and the 21st Century Skills Frameworks and PBLs in STEM

The third session will outline several 21st century skill frameworks and teaching approaches and how mathematics educators, researchers and practitioners can find a synergistic way to bring important process skills without overwhelming teachers and students. We will discuss the ways elementary teachers can make connections between the problem-based ways they have engaged students in mathematical modeling and STEM. The teachers are able to take advantage of interdisciplinary opportunities across the subjects they teach and find complementary connections between subjects and common classroom practices that support MM.

Anticipated Follow-up Activities and Goals of Working Group

In the spirit of exploring the theme of the Synergy at the Crossroads: Future Directions for Theory, Research, and Practice, each session will engage participants to share their research interests related to mathematical modeling and form groups that might pursue research collaboratively based on the interests of the participants. Some of the questions include:

• What defines successful mathematical modeling lessons at different grade levels?
• What can we learn from teachers who implement MM regularly in their classrooms?
• How can we support teachers enacting MM through lesson plans and other resources?
• How can we map out the learning progression of MM across grade levels?
• How and what can we learn about models elicited from student artifacts from MM tasks?
• What do “successful” modeling practices look like in our elementary mathematics classrooms? How are they similar or different from practices in secondary classrooms?
• What does it mean to “see the math” in the components of mathematical modeling?
• How do teachers select and/or develop modeling problems? How can PLCs or Teacher Study Groups help teachers anticipating how students will answer the MM questions?

Our goal is for the working group leaders to propose an edited handbook or a special issues journal venue for mathematical modeling where participants interested in submitting manuscripts can work together to provide a comprehensive research trajectory documenting the progression of mathematical modeling from emergent levels to more sophisticated levels of modeling.

Acknowledgments

This working group was funded by NSF 1441024-STEM C

References

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IMMERSION funded by NSF STEM-C 1440124